64.18.3 problem 23

Internal problem ID [13555]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.4. Exercises page 309
Problem number : 23
Date solved : Monday, March 31, 2025 at 08:01:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+7 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 9\\ y \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 33

ode:=[diff(x(t),t) = -2*x(t)+7*y(t), diff(y(t),t) = 3*x(t)+2*y(t)]; 
ic:=x(0) = 9y(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{5 t}+7 \,{\mathrm e}^{-5 t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{5 t}-3 \,{\mathrm e}^{-5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 38

ode={D[x[t],t]==-2*x[t]+7*y[t],D[y[t],t]==3*x[t]+2*y[t]}; 
ic={x[0]==9,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 7 e^{-5 t}+2 e^{5 t} \\ y(t)\to 2 e^{5 t}-3 e^{-5 t} \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - 7*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {7 C_{1} e^{- 5 t}}{3} + C_{2} e^{5 t}, \ y{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{5 t}\right ] \]